a(n) is the Sprague-Grundy value of the Node-Kayles game on the semi-regular graph of n linked 4-cycles with vertex set {u_1, u_2, ..., u_n, u_{n+1}, v_1, w_1, v_2, w_2, ..., v_n, w_n}. In this graph, u_1, u_2, ..., u_n, and u_{n+1} form a path, and additional edges are given by {u_i, v_i}, {v_i, w_i}, and {w_i, u_{i+1}} for all i=1,2,...,n.

A similar graph is given by n linked 4-cycles with vertex set {u_1, u_2, ..., u_n, u_{n+1}, v_1, w_1, v_2, w_2, ..., v_n, w_n}. In this graph, edges are given by {u_i, v_i}, {u_i, w_i}, {v_i, u_{i+1}}, and {w_i, u_{i+1}} for all i=1,2,...,n. The Sprague-Grundy value of the Node-Kayles game played on this graph is 0 if n is odd and 1 otherwise.

We define b(n) and c(n), as well as their recurrence relations, to be used in the recurrence relation for a(n).

Let b(n) be the Sprague-Grundy value of the Node-Kayles game on the graph of n linked 4-cycles with vertex set {u_1, u_2, ..., u_n, u_{n+1}, u_{n+2}, u_{n+3}, v_1, w_1, v_2, w_2, ..., v_n, w_n}. In this graph, u_1, u_2, ..., u_n, u_{n+1}, u_{n+2}, and u_{n+3} form a path, and additional edges are given by {u_i, v_i}, {v_i, w_i}, and {w_i, u_{i+1}} for all i=1,2,...,n.

Let c(n) be the Sprague-Grundy value of the Node-Kayles game on the graph of n linked 4-cycles with vertex set {u_{-1}, u_0, u_1, u_2, ..., u_n, u_{n+1}, u_{n+2}, u_{n+3}, v_1, w_1, v_2, w_2, ..., v_n, w_n}. In this graph, u_{-1}, u_0, u_1, u_2, ..., u_n, u_{n+1}, u_{n+2}, and u_{n+3} form a path, and additional edges are given by {u_i, v_i}, {v_i, w_i}, and {w_i, u_{i+1}} for all i=1,2,...,n.

In the following recurrence relations, '+' is the bitwise XOR operator.